The Crouzeix constant
Description of constant
$C_{2}$ is the Crouzeix constant (sometimes denoted $Q$). It is the smallest constant $C$ such that for every $n \ge 1$, every complex matrix $A \in \mathbb{C}^{n \times n}$, and every complex polynomial $p$ one has
\[\|p(A)\| \ \le\ C \ \max_{z \in W(A)} |p(z)|,\]where $|\cdot|$ is the operator norm induced by the Euclidean norm (i.e. the spectral norm), and
\[W(A) := \{ v^\ast A v : v \in \mathbb{C}^n,\ \|v\|_2 = 1\}\]is the numerical range (field of values) of $A$.
Equivalently,
\[C_{2} = \sup_{n \ge 1}\ \sup_{A \in \mathbb{C}^{n\times n}}\ \sup_{p \not\equiv 0} \frac{\|p(A)\|}{\max_{z \in W(A)} |p(z)|}.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $11.08$ | [C2007] | First dimension-free bound. Also holds in the completely bounded (matrix-valued) setting. |
| $1+\sqrt{2} \approx 2.41421$ | [CP2017] | Best universal upper bound prior to the [Jin2026] / [LS2026] pair. Also holds in the completely bounded setting. |
| $2$ | [Jin2026] | Jin: claimed proof of Crouzeix’s conjecture, posted on preprints.org in July 2026. |
| $2$ | [LS2026] | Lorist–Schwenninger: proof of Crouzeix’s conjecture, combining previous tools for weaker estimates with a “simple perturbation lemma for 2-dilations” applied to the iterates $f^n$ in the double-layer potential representation. The paper acknowledges [Jin2026]. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial | Take $p \equiv 1$. |
| $2$ | [C2007] | Achieved by $p(z)=z$ and $A=\begin{pmatrix}0 & 2\\ 0 & 0\end{pmatrix}$, for which $W(A)$ is the unit disk. |
Additional comments and links
- Crouzeix conjectured (in [C2004]) that $C_{2}=2$. The lower bound $2$ matches the upper bounds of [Jin2026] and [LS2026], so the conjecture is (claimed to be) settled: $C_{2} = 2$. Jin’s preprint appeared in July 2026 and Lorist–Schwenninger’s followed in August 2026, acknowledging Jin’s.
- The conjectured constant $2$ is known to hold in a number of special cases; see, for instance, [Cho2013], [GKL2018], [CGL2018].
- Numerical experiments strongly support the conjecture; see [GO2018].
- Wikipedia page on Crouzeix’s conjecture
- AIM workshop page on Crouzeix’s conjecture
References
- [C2004] Crouzeix, Michel. Bounds for analytical functions of matrices. Integral Equations and Operator Theory 48 (2004), no. 4, 461–477. DOI: 10.1007/s00020-002-1188-6.
- [C2007] Crouzeix, Michel. Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244 (2007), no. 2, 668–690. DOI: 10.1016/j.jfa.2006.10.013.
- [CP2017] Crouzeix, Michel; Palencia, César. The Numerical Range is a $(1+\sqrt2)$-Spectral Set. SIAM J. Matrix Anal. Appl. 38 (2017), no. 2, 649–655. DOI: 10.1137/17M1116672.
- [DD1999] Delyon, Bernard; Delyon, François. Generalization of Von Neumann’s spectral sets and integral representation of operators. Bull. Soc. Math. France 127 (1999), 25–42. (See also: https://www.numdam.org/article/BSMF_1999__127_1_25_0.pdf)
- [Cho2013] Choi, Daeshik. A proof of Crouzeix’s conjecture for a class of matrices. Linear Algebra Appl. 438 (2013), no. 8, 3247–3257. DOI: 10.1016/j.laa.2012.12.045.
- [GO2018] Greenbaum, Anne; Overton, Michael L. Numerical investigation of Crouzeix’s conjecture. Linear Algebra Appl. 542 (2018), 225–245. DOI: 10.1016/j.laa.2017.04.035.
- [GKL2018] Glader, Christer; Kurula, Mikael; Lindström, Mikael. Crouzeix’s conjecture holds for tridiagonal $3\times 3$ matrices with elliptic numerical range centered at an eigenvalue. SIAM J. Matrix Anal. Appl. 39 (2018), no. 1, 346–364. DOI: 10.1137/17M1110663. arXiv:1701.01365.
- [CGL2018] Caldwell, Trevor; Greenbaum, Anne; Li, Kenan. Some Extensions of the Crouzeix–Palencia Result. SIAM J. Matrix Anal. Appl. 39 (2018). DOI: 10.1137/17M1140832. arXiv:1707.08603.
- [RS2018] Ransford, Thomas; Schwenninger, Felix L. Remarks on the Crouzeix–Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set. SIAM J. Matrix Anal. Appl. 39 (2018), no. 1, 342–345. DOI: 10.1137/17M1143757. arXiv:1708.08633.
- [Jin2026] Jin. A proof of Crouzeix’s conjecture. Preprints.org manuscript 202607.1919 (July 2026).
- [LS2026] Lorist, Emiel; Schwenninger, Felix L. A solution to Crouzeix’s conjecture. arXiv:2608.03841 (2026).
Acknowledgements
Prepared with ChatGPT 5.2 Pro.